From a Scalar Parabolic Oscillator to Topological Thermostats: Selective Feeback Control of Harmonic Flow Modes

What the math gives to ML

The paper gives a constructive orthogonal decomposition of graph edge flows into a cut component, determined by node potentials, and a harmonic cycle component, invisible to the incidence divergence operator. Its transferable asset is selective control: one can contract or regulate cycle modes without disturbing the node-relevant cut dynamics, while explicitly measuring the spillover caused by localized channels. This suggests Hodge-aware graph layers and recurrent graph dynamics that separately process divergence-carrying information and circulation information instead of allowing message passing to mix them implicitly. The first implementation should use exact sparse graph projections on small and medium graphs, then compare stability, long-horizon rollout error, and parameter efficiency against ordinary edge-GNN residual blocks.